Relative Calabi-Yau structures and ice quivers with potential

In 2015, Van den Bergh showed that complete $3$-Calabi-Yau algebras over an algebraically closed field of characteristic $0$ are equivalent to Ginzburg dg algebras associated with quivers with potential.

He also proved the natural generalisation to higher dimensions and non-algebraically closed ground fields.

The relative version of the notion of Ginzburg dg algebra is that of Ginzburg morphism.

For example, every ice quiver with potential gives rise to a Ginzburg morphism.

We generalise Van den Bergh's theorem by showing that, under suitable assumptions, any morphism with a relative Calabi-Yau structure is equivalent to a Ginzburg(-Lazaroiu) morphism.

In particular, in dimension $3$ and over an algebraically closed ground field of characteristic $0$, it is given by an ice quiver with potential.

Thanks to the work of Bozec-Calaque-Scherotzke, this result can also be viewed as a noncommutative analogue of Joyce-Safronov's Lagrangian neighbourhood theorem in derived symplectic geometry.